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Fungrim entry: 21f4f9

dndzn[ψ(m) ⁣(z)]=ψ(m+n) ⁣(z)\frac{d^{n}}{{d z}^{n}} \left[\psi^{(m)}\!\left(z\right)\right] = \psi^{(m + n)}\!\left(z\right)
Assumptions:nZ0  and  mZ0  and  zC  and  z{0,1,}n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}
\frac{d^{n}}{{d z}^{n}} \left[\psi^{(m)}\!\left(z\right)\right] = \psi^{(m + n)}\!\left(z\right)

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}
Fungrim symbol Notation Short description
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
DigammaFunctionψ ⁣(z)\psi\!\left(z\right) Digamma function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
    Formula(Equal(ComplexDerivative(Brackets(DigammaFunction(z, m)), For(z, z, n)), DigammaFunction(z, Add(m, n)))),
    Variables(n, m, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(m, ZZGreaterEqual(0)), Element(z, CC), NotElement(z, ZZLessEqual(0)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC